C₄₇×#10624751 #46804392

order	34263
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃³×C₄₇ · Schur multiplier (H₂) = C₃⁵

Aliases

direct product

C₄₇×#10624751

Subgroups

	subgroup	G/H
center, socle	C₃×C₄₇	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₄₇
prime radical	C₄₇×#6911	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₄₇ ◁ C₃³×C₄₇ ◁ G

414 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 3807

A minimal faithful permutation representation — 3 generators on 128 points:

(1,2,6)(3,11,20)(4,8,21)(5,10,22)(7,23,15)(9,29,19)(12,33,43)(13,36,49)(14,26,45)(16,27,46)(17,30,54)(18,28,47)(24,50,40)(25,44,38)(31,48,35)(32,57,39)(34,56,72)(37,52,74)(41,53,69)(42,55,68)(51,66,62)(58,71,65)(59,70,61)(60,73,80)(63,67,64)(75,79,78)(76,81,77)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128);(1,3,12,5,15,35,19,14,40)(2,7,24,10,26,33,9,11,31)(4,13,34,18,39,62,17,38,61)(6,20,43,22,23,48,29,45,50)(8,25,51,28,36,59,30,32,56)(16,37,60,41,64,77,42,65,78)(21,44,66,47,49,70,54,57,72)(27,52,73,53,63,76,55,58,75)(46,67,79,69,71,80,68,74,81);(1,4,16,5,18,41,19,17,42)(2,8,27,10,28,53,9,30,55)(3,13,37,15,39,64,14,38,65)(6,21,46,22,47,69,29,54,68)(7,25,52,26,36,63,11,32,58)(12,34,60,35,62,77,40,61,78)(20,44,67,23,49,71,45,57,74)(24,51,73,33,59,76,31,56,75)(43,66,79,48,70,80,50,72,81)

Conjugacy classes & character table

2679 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34263, 311);